Disorder and superfluid density in overdoped cuprate superconductors
DDisorder and superfluid density in overdoped cuprate superconductors
N. R. Lee-Hone, J. S. Dodge,
1, 2 and D. M. Broun
1, 2 Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada Canadian Institute for Advanced Research, Toronto, ON, MG5 1Z8, Canada
We calculate superfluid density for a dirty d -wave superconductor. The effects of impurity scat-tering are treated within the self-consistent t -matrix approximation, in weak-coupling BCS theory.Working from a realistic tight-binding parameterization of the Fermi surface, we find a superfluiddensity that is both correlated with T c and linear in temperature, in good correspondence withrecent experiments on overdoped La − x Sr x CuO . PACS numbers: 74.25.Ha,74.20.Fg,74.20.Rp,74.72.Gh, 74.62.En
I. INTRODUCTION
The superfluid density, ρ s , plays a special role in thephysics of cuprate superconductors, as it determines thestiffness of the superconducting order parameter to fluc-tuations in its phase. In most superconductors ρ s islarge: phase fluctuations are heavily suppressed and thetransition temperature T c is set primarily by the gap en-ergy 2∆ required to break a Cooper pair. Cuprates,on the other hand, have a relatively low carrier den-sity, which limits ρ s and leaves them susceptible to phasefluctuations. As a result, both ∆ and ρ s may influence T c in the cuprates, with phase fluctuations playing anincreasingly dominant role as ρ s approaches zero. Ex-periments support this view: in hole-doped cuprates inthe underdoped regime, T c correlates closely with ρ s , whereas there is a large energy gap that extends wellinto the normal state. In addition, a variety of super-conducting fluctuation effects have been observed above T c in underdoped cuprates. On the overdoped sidethe situation is different: T c appears to closely track theenergy gap, as it would in a conventional super-conductor; and, while the correlation between T c and ρ s remains, the causal relationship between these quan-tities is far less clear. Complicating the chain of causalityis another parameter — disorder — that directly influ-ences ρ s , T c , and ∆ in a d -wave superconductor. Oneof our main purposes in this paper is to explore the extentto which disorder is an important driver of the relation-ships between the other three quantities, with particularattention to the case of the overdoped cuprates.Strong motivation comes from a recent experiment, which provides exhaustive evidence that the superfluidstiffness and the superconducting transition temperatureof overdoped La − x Sr x CuO approach zero in tandemas a function of doping. The new study reinforces theargument that the close correlation between T c and ρ s is a significant and intrinsic feature of the overdopedcuprates. It also shows that the superfluid density re-tains an approximately linear temperature dependenceover a wide temperature range, as expected for a d -wavesuperconductor in the clean limit. Together, theseobservations present a puzzle: on the one hand the su-perfluid stiffness is expected to correlate with T c in the dirty limit, because the normal-state spectral weight iscut off by the gap, but on the other hand the ob-served temperature dependence of ρ s appears to excludethis possibility.To try to resolve this contradiction we have revisitedthe theoretical relationship between disorder, superfluiddensity and T c within dirty d -wave BCS theory. In adirty d -wave superconductor it is well known that strong-scattering (unitarity-limit) impurities rapidly induce acrossover from T -linear superfluid density to quadraticbehaviour below a crossover temperature T ∗ thatis proportional to the geometric mean of the normal-state impurity scattering rate and the superconductingenergy gap. The corresponding loss of superfluid den-sity is of order T ∗ /T c . Therefore, in this limit, anysignificant loss of superfluid density must be accompa-nied by a very visible crossover to quadratic behaviourin ρ s ( T ). This result is so well known that the argu-ment is frequently run in reverse, with the measuredvalue of T ∗ used to place an upper bound on the de-gree of superfluid suppression and T c suppression due toimpurities. In fact, we will show that the reverse argu-ment breaks down for weak-scattering (Born-limit) disor-der, and approximately linear-in-temperature superfluiddensity can coexist with substantial suppression of su-perfluid density and T c . We have carried out calcula-tions of superfluid density for realistic, doping-dependentFermi surfaces based on tight-binding parameterizationsof angle-resolved photoemission (ARPES) dispersions forLa − x Sr x CuO . This turns out to be crucial to car-rying out a detailed comparison with ρ s ( T ) data onLa − x Sr x CuO . In the calculations, the effects of dis-order on the quasiparticle energies and lifetimes, and onthe superconducting energy gap and T c , are calculatedusing the self-consistent t -matrix approximation, withinthe weak-coupling limit of d -wave BCS theory. Weconclude that it is possible to obtain a superfluid densitythat is both correlated with T c and linear in temperature. a r X i v : . [ c ond - m a t . s up r- c on ] J un II. THEORYA. Dirty d-wave superconductivity
The gap equation for a weak-coupling d -wave super-conductor can be written in the imaginary-axis formalismas ∆ k = 2 πT ω (cid:88) ω n > (cid:42) V k , k (cid:48) ∆ k (cid:48) (cid:112) ˜ ω n + ∆ k (cid:48) (cid:43) FS , (1)where ∆ k is the gap parameter at wave-vector k , ω n = 2 πT ( n + ) are the fermionic Matsubara frequen-cies, V k , k (cid:48) is the pairing interaction, ω is a high fre-quency cut off, and (cid:104) ... (cid:105) FS denotes an average over theFermi surface.In the self-consistent t -matrix approximation, point-like, nonmagnetic impurities renormalize thefermionic Matsubara frequencies according to˜ ω n ≡ ˜ ω ( ω n ) = ω n + π Γ (cid:104) N k (˜ ω n ) (cid:105) FS c + (cid:104) N k (˜ ω n ) (cid:105) . (2)Here c is the cotangent of the scattering phase shift, Γ isa scattering parameter proportional to the concentrationof impurities and N k (˜ ω n ) = ˜ ω n (cid:112) ˜ ω n + ∆ k . (3)For a d -wave order parameter, which averages to zeroover the Fermi surface, there is no explicit impurity renor-malization of ∆ k , just an indirect reduction through theeffect of the impurities on ˜ ω n .For simplicity, we assume a separable pairing interac-tion based on a d -wave form factor Ω k defined in the firstBrillouin zone of the two-dimensional CuO planes,Ω k ∝ (cid:0) cos( k x a ) − cos( k y a ) (cid:1) , (4)where a is the lattice spacing and Ω k is normalized suchthat (cid:104) Ω k (cid:105) FS = 1. The pairing interaction therefore takesthe form V k , k (cid:48) = V Ω k Ω k (cid:48) . (5)We will see below that for weak-coupling BCS, wherethe cut-off frequency of the interaction, ω , is much largerthan the superconducting transition temperature, T c , thecombined effect of V and ω is captured by the clean-limit transition temperature, T c , so that V and ω donot appear explicitly as parameters in the theory. Intro-ducing a temperature-dependent gap amplitude, ψ ( T ),the gap equation becomes∆ k ≡ ψ Ω k = 2 πT ω (cid:88) ω n > (cid:42) V Ω k Ω k (cid:48) ψ Ω k (cid:48) (˜ ω n + ψ Ω k (cid:48) ) (cid:43) FS . (6) Cancelling common factors, rearranging and reassign-ing k (cid:48) → k we have1 V = 2 πT ω (cid:88) ω n > (cid:42) Ω k (˜ ω n + ψ Ω k ) (cid:43) FS . (7)In the absence of disorder the quasiparticle energies areunrenormalized (˜ ω n = ω n ) and the gap vanishes ( ψ → T c . Using (cid:104) Ω k (cid:105) FS = 1, we have at this temperature1 V = 2 πT c ω (cid:88) ω n > ω n ( T c ) ≈ ln (cid:18) ω . T c (cid:19) , (8)where the approximation is valid when ω (cid:29) T c . Thisrearranges to give the familiar weak-coupling BCS result T c = 1 . ω exp( − /V ) . (9)The logarithmic temperature dependence in Eq. 8 can beused to obtain an expression for the coupling constant V that applies at any arbitrary temperature T : V = 2 πT ω (cid:88) ω n > ω n ( T ) + ln (cid:18) TT c (cid:19) . (10)This allows V to be eliminated from the gap equation,which then takes the formln (cid:18) T c T (cid:19) = 2 πT ∞ (cid:88) ω n > (cid:32) ω n − (cid:42) Ω k (˜ ω n + ψ Ω k ) (cid:43) FS (cid:33) . (11)Rapid convergence lets the Matsubara sum to be takento infinity, eliminating explicit dependence on ω . For agiven choice of Fermi surface and impurity parameters,Eqs. 2 and 11 are solved self consistently to obtain ˜ ω n ( T )and ψ ( T ).In the presence of disorder the energy gap closes ata reduced transition temperature, T c . For T ≥ T c , N k (˜ ω ) → t -matrix equation describing the im-purity scattering, Eq. 2, simplifies to˜ ω ( ω n ) = ω n + π Γ1 + c ≡ ω n + Γ N . (12)The imaginary part of the self energy in this limit is de-noted Γ N , the normal-state scattering rate due to im-purities. Equation 11 can be solved with ψ → ω n → ω n + Γ N to determine T c :ln (cid:18) T c T c (cid:19) = 2 πT c ∞ (cid:88) ω n > (cid:18) ω n − ω n + Γ N (cid:19) (13)= ∞ (cid:88) ω n > (cid:32) n + − n + + Γ N πT c (cid:33) (14)= ψ (cid:18) + Γ N πT c (cid:19) − ψ (cid:0) (cid:1) , (15)where ψ ( x ) is the digamma function. B. Superfluid density
The zero-temperature, zero-disorder penetrationdepth, λ , is closely related to the bare plasmafrequency, ω p . The corresponding superfluid density is ρ s ≡ λ = µ (cid:15) ω p (16)= 2 µ e (cid:90) + πd − πd d k z π (cid:90) d k (2 π ) δ ( (cid:15) F − (cid:15) k ) v k ,x . (17)Here we specialize to a quasi-2D material with layerspacing d and in-plane energy dispersion (cid:15) k . (cid:15) F is theFermi energy, k = ( k x , k y ) is the in-plane momentum and v k = h (cid:0) ∂∂k x , ∂∂k y (cid:1) (cid:15) k is the in-plane velocity. We changeEq. 17 to a Fermi surface integral by transforming coor-dinates from ( k x , k y ) to ( (cid:15), φ ), where φ is the angle in theplane, measured about (cid:0) πa , πa (cid:1) at low hole dopings and(0 ,
0) at higher dopings. The Jacobian of the transfor-mation is J ( φ ) = ∂ ( k x , k y ) ∂ ( (cid:15), φ ) = | k | ¯ h k · v k . (18)When the energy and k z integrations are carried out weobtain 1 λ = µ e π ¯ hd (cid:90) π | k F | k F · v F v F,x d φ , (19)where the Fermi wavevector k F and Fermi velocity v F are functions of φ . The Fermi surface average used in theprevious section, (cid:104) ... (cid:105) FS , must include the same Jacobianfactor: (cid:10) A ( φ ) (cid:11) FS ≡ (cid:90) π J ( φ ) A ( φ )d φ (cid:46)(cid:90) π J ( φ )d φ . (20)For calculation of plasma frequency and superfluid den-sity we define a second Fermi surface average, (cid:104)(cid:104) ... (cid:105)(cid:105) FS ,that contains the additional factor of v F,x : (cid:10)(cid:10) A ( φ ) (cid:11)(cid:11) FS ≡ (cid:90) π J ( φ ) A ( φ ) v F,x d φ (cid:46)(cid:90) π J ( φ ) v F,x d φ . (21)For Fermi surfaces that are close to circular these dis-tinctions are usually not important. However, for theoverdoped cuprates, the details of the Fermi-surface av-erages turn out to be crucial to understanding the tem-perature dependence of superfluid density measured inexperiments, and so are given here in full.The finite temperature superfluid density, in the pres-ence of disorder, is most efficiently calculated using aMatsubara sum. Normalized to ρ s it is given by ρ s ( T ) ρ s = 2 πT ∞ (cid:88) ω n > (cid:42)(cid:42) ∆ k (˜ ω n + ∆ k ) (cid:43)(cid:43) FS , (22)where the effects of disorder are built in via the renor-malized Matsubara frequencies and gap. T / T c0 ρ s / ρ s G = . T c N G = . T c N G = . T c N G = N Born limitUnitarity limitClean d -wave1.0 FIG. 1. Normalized superfluid density, ρ s /ρ s , for a d -wavesuperconductor with a circular Fermi surface. The degree ofscattering is characterized by the normal-state scattering rate,Γ N , in units of the clean limit transition temperature, T c , forscatterers acting in the Born limit ( c (cid:29)
1) and the unitaritylimit ( c = 0). The temperature dependence of the gap, ∆( T ),has been calculated self-consistently for each set of impurityparameters, assuming a separable d -wave pairing interaction. C. Impurity contribution to normal-stateresistivity
We use the normal-state impurity scattering rate Γ N toparameterize the amount of scattering in the theory. Mo-tivated by the known types of elastic-scattering disorderin La − x Sr x CuO and other cuprates, we allow for twotypes of defects acting in combination: weak-limit scat-terers, parameterized by Γ N, Born , to capture the effectof out-of-plane defects such as Sr dopants; and strong-scattering disorder, parameterized by Γ N, unitarity , to rep-resent native defects in the CuO planes, such as Cu va-cancies. The combined effect of Born and unitarity-limitscattering is additive in the self energy (Eq. 2). An estimate of the scattering parameters to be used inthe model can be made by comparing Γ N with experi-ment, taking care to note that the experimentally acces-sible scattering rate ( e.g. , that observed in an ARPESmeasurement of inverse lifetime ) is 2Γ N . Keeping thisin mind and assuming for now that the momentum re-laxation rate is the same as the single-particle scatteringrate, the dc resistivity due to impurity scattering will be ρ = 2Γ N (cid:15) ω p = µ λ × N . (23)Here λ is the zero-temperature penetration depth ofa notional system with the same Fermi surface (dopinglevel) that does not contain disorder. It cannot be ac-cessed experimentally but an estimate of λ can be made (cid:31)Π(cid:29)Π(cid:28)(cid:31)(cid:27)(cid:29)(cid:27)(cid:28) - t ’/ t = 0.150 Ε / t = 0.810(a) p = 0.16 (cid:31)Π(cid:29)Π(cid:28)(cid:31)(cid:27)(cid:29)(cid:27)(cid:28) - t ’/ t = 0.142 Ε / t = 0.832(b) p = 0.185 (cid:31)Π(cid:29)Π(cid:28)(cid:31)(cid:27)(cid:29)(cid:27)(cid:28) - t ’/ t = 0.140 Ε / t = 0.838(c) p = 0.1914 (cid:31)Π(cid:29)Π(cid:28)(cid:31)(cid:27)(cid:29)(cid:27)(cid:28) - t ’/ t = 0.135 Ε / t = 0.858(d) p = 0.21 (cid:31)Π(cid:29)Π(cid:28)(cid:31)(cid:27)(cid:29)(cid:27)(cid:28) - t ’/ t = 0.125 Ε / t = 0.918(e) p = 0.26 FIG. 2. (color online) Constant energy contours in momentum space for optimally to overdoped La − x Sr x CuO , at selectednominal hole-dopings p , based on a tight-binding parameterization of ARPES spectra. Momentum is measured in units ofinverse lattice spacing, 1 /a . Fermi surfaces are depicted by solid black lines. Doping-dependent tight-binding parameters t (cid:48) and (cid:15) are indicated on the plots, in units of nearest-neighbour hopping integral t . starting from the measured zero-temperature penetrationdepth, λ , and then correcting for the degree of superfluidsuppression using the T → λ = λ (cid:46) ρ s ( T → ρ s . (24)The final form for the residual resistivity is then ρ = 2 µ λ Γ N (cid:46) ρ s ( T → ρ s . (25)We note that this neglects the effects of small-angle scat-tering, making it an upper bound on resistivity. If known,the small-angle scattering correction can be applied toEq. 25 as a refinement. III. COMPARISON WITH EXPERIMENTA. Isotropic systems
Sufficiently far from half-filling the Fermi surface of aquasi-2D metal is well approximated by a circle, and the d -wave form factor is Ω( φ ) ≈ √ φ ). In this limitthe angle integrals can be evaluated analytically andthe Matsubara sums computed rapidly. Results for thesuperfluid density are shown in Fig. 1. The clean-limitcurve displays one of the clear hallmarks of d -wave gapnodes: linear behaviour in ρ s ( T ). Note that this be-haviour emerges only in the asymptotic low-temperaturelimit — the substantial downwards curvature in ρ s ( T ) athigher temperatures is a band-structure effect due, in thiscase, to the particular choice of a circular Fermi surface.As discussed above, it is convenient to parameterizethe disorder level in terms of normal-state scatteringrate Γ N . We see in Fig. 1 that while T c depends onlyon Γ N /T c , the form of ρ s ( T ) at lower temperatures isstrongly affected by the impurity phase shift. In both theBorn ( c (cid:29)
1) and unitarity ( c = 0) limits there is sub-stantial suppression of the zero-temperature superfluiddensity but it is only in the unitarity limit that disor-der rapidly causes a cross-over to quadratic behaviour in ρ s ( T ) at low temperatures. In contrast, it takes alarge amount of Born scattering (and subsequent loss ofsuperfluid density) before the low temperature linear be-haviour in ρ s ( T ) is removed. Figure 1 therefore serves toillustrate that while the observation of T behaviour in ρ s ( T ) is a concrete indication that disorder is important,the observation of a linear temperature dependence of ρ s does not guarantee that a material is a clean d -wavesuperconductor. B. Overdoped cuprates
The mid-range curvature of ρ s ( T ) seen in Fig. 1is typically not observed in overdoped cupratesuperconductors. To carry out a more detailedcomparison with the experiments on La − x Sr x CuO requires realistic Fermi surfaces. The calculations pre-sented below are based on next-next-nearest neighbourtight-binding parameterizations of (cid:15) k in La − x Sr x CuO , (cid:15) k = (cid:15) − t (cos k x a + cos k y a ) − t (cid:48) cos k x a cos k y a − t (cid:48)(cid:48) (cos 2 k x a + cos 2 k y a ) , (26)obtained from fits to ARPES spectra as a function ofhole doping. In the ARPES study t = 0 .
25 eV and t (cid:48)(cid:48) /t (cid:48) = − . t (cid:48) and (cid:15) are parameters that vary withdoping, leading to the energy dispersions and Fermi sur-faces shown in Fig. 2. In this model the Fermi surface isdefined by (cid:15) k = 0.To bridge between superfluid density and ARPES mea-surements we assume the standard parabolic relationshipbetween T c and hole doping p . The specific form used inRef. 30 is p = 0 .
16 + (0 . − . × − T c ) / . (27)This maps fairly closely onto the stated Sr concentrationsin the ARPES experiment of Ref. 40, with a slight offset: i.e. , x = 0 . → p = 0 .
16 and x = 0 . → p = 0 .
23. Infuture, it would be highly informative if ARPES mea-surements could be carried out on samples similar tothose used in the penetration depth measurements, so T (K) ρ s ( K ) ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ρ s0 (K) T c ( K ) T (K) ρ s / ρ s T c ( K ) a b M F op t ρ s0 / ρ s00opt T c MF T c FIG. 3. (Color online) Superfluid density in overdoped La − x Sr x CuO . (a) Main panel: Superfluid density for overdopedLa − x Sr x CuO from Ref. 30, replotted in two dimensions, over the full temperature range that the data were measured: T >
300 mK for the lowest T c sample, T > . T c samples. Shading indicates 2% error bands in superfluid density(1% error bands in penetration depth) as stated in Ref. 30. The dashed line shows an example of the construction used toestimate the transition temperature, T MF c , to be assumed in the corresponding mean-field theory. Inset: correlation betweenmeasured superconducting transition temperature, T c , and zero-temperature superfluid density, ρ s , replotted from Ref. 30.(b) Superfluid density calculated within dirty d -wave BCS theory, on ARPES-derived Fermi surfaces, for predominantly weakscattering (Γ N, Born = 17 K) with a small amount of strong-scattering disorder (Γ N, unitarity = 1 K). Main panel: Superfluiddensity ρ s normalized to the zero-temperature, clean-limit superfluid density for optimally doped material, ρ opt s . Solid linescorrespond to the temperature range accessed in the experiments, dashed lines extend this to lower temperatures. Inset:Correlation between mean-field transition temperature, T MF c , and ρ s /ρ opt s . that details in the electronic structure could be lined upprecisely with the doping-dependent superfluid density.In any case, we note that the parameter p is a nominalhole doping, and is only used in the calculations as aninternal variable. The doping dependence of the calcu-lated superfluid density shown in Fig. 3 is not sensitiveto the detailed mapping onto the ARPES experiment. Inparticular, the change in Fermi surface topology as theFermi energy passes through the van Hove point does notappear as a sharp feature in ρ s ( p ); this is due to the fac-tor of v F,x in the relevant Fermi surface integral, whichunderweights the parts of the Fermi surface where thedispersion is flat. The only sign of the van Hove cross-ing is a small cusp in the zero-temperature gap ratio,2∆ /k B T c , as can be seen in the inset of Fig. 4. In thecurrent context, the most important consequence of bas-ing the calculations on realistic energy dispersions is theremoval of spurious midrange curvature in ρ s ( T ). C. Disorder level
In deciding on the appropriate level of disorder to as-sume in the calculations, useful guidance comes from dc resistivity. As discussed in Sec. II, the scattering raterelevant to pair-breaking and superfluid density is theelastic scattering rate, which is related to the residualresistivity ρ ≡ ρ ( T → T c . Toavoid uncertainties associated with extrapolating resis-tivity below T c , we assume a single, doping-independentscattering rate in our calculation, as our primary aim isto illustrate the physics contained within dirty d -wavesuperconductivity. In addition, as mentioned previously,the theoretical resistivity given in Eqs. 23 and 25 does notaccount for the effects of small-angle scattering, which inthe cuprates can cause the momentum relaxation rate,Γ tr , to be substantially smaller than the single-particlerelaxation rate, Γ sp . Overdoped Tl Ba CuO δ pro-vides a useful point of reference here, as it is sufficientlyclean that quantum oscillation experiments have beenperformed. This enables the single-particle relax-ation rate and momentum relaxation rate to be de-termined separately, with the result that Γ sp / Γ tr = 1 . In the absence of quantum oscillation data, we proceedby assuming the same ratio for La − x Sr x CuO . Thisis not unreasonable, as the dominant source of disor-der in both systems is out-of-plane cation disorder: inthe case of Tl Ba CuO δ , an excess of Cu atoms, oc-
10 20 30 40 50020406080100120 T (K) Δ / k B ( K ) T c (K) Δ / k B T c FIG. 4. (Color online) Superconducting gap parameter under-lying the superfluid density calculation. Main panel: Tem-perature and doping dependence of the gap maximum onthe Fermi surface. Inset: Doping dependence of the zero-temperature gap ratio 2∆ /k B T c . Dashed line denotes theclean-limit d -wave BCS value, 2∆ = 4 . k B T c . cupying Tl sites; and, in La − x Sr x CuO , the partialsubstitution of Sr for La that is an inherent part of itshole-doping mechanism. With these assumptions, andtaking a residual resistivity ρ ≈ µ Ω cm, we obtaina normal-state scattering rate Γ N ≈
18 K. In the cal-culations presented here we partition this between pre-dominantly weak, Born-limit scattering (Γ N, Born = 17 K)with a small amount of strong, unitarity-limit scattering(Γ N, unitarity = 1 K). The inclusion of a small amountof strong-scattering disorder causes a low-temperaturecrossover to T dependence in ρ s ( T ), which is hinted atby the data from Ref. 30, plotted in Fig. 3(a). Note thatthis choice of scattering rate satisfies the conditions forclean-limit superconductivity, Γ tr (cid:28) , over most ofthe doping range. In fact, based on the scattering rateestimates above and the gap values plotted in Fig. 4, weestimate Γ tr ∼ only for T c < d -wave superconductorsbeing a process that spreads uncondensed spectral weightover a wide range of sub-gap frequencies. IV. DISCUSSION
The calculated superfluid density is presented inFig. 3(b). In our calculations the underlying Fermisurface follows the doping dependence of the electronicstructure measured in the ARPES experiments and isnot an adjustable parameter. At each value of the nom- T ( K ) - t ' / t p ϵ / t T c T cc T MF abc FIG. 5. (Color online) Doping-dependence of model pa-rameters. (a) Clean-limit transition temperature, T c andmean-field transition temperature, T MF c , plotted along withthe experimentally observed transition temperature, T c , fromRef. 30. (b) Next-nearest neighbour hopping integral, t (cid:48) , inunits of the nearest neighbour hopping, t . (c) Energy offset, (cid:15) , in units of t . Doping-dependent tight-binding parametersare interpolations through discrete values (solid points) fromRef. 40. inal hole doping p , the underlying clean-limit transitiontemperature, T c , is set according to Eq. 15 so that T c ,the transition temperature in the presence of disorder,matches the mean-field transition temperature, T MF c , in-ferred by extrapolating the linear regime of the experi-mental ρ s ( T ) curve to zero, as shown in Fig. 3(a). Thedoping dependences of T c , T MF c and T c are plotted inFig. 5(a) and the tight-binding parameters used in thecalculation are shown in Figs. 5(b) and 5(c). As describedin the previous section, the amount of impurity scatter-ing is the only independent parameter in the theory andhas been fixed as a function of doping, for the sake ofsimplicity, as discussed in Sec. III C.The superfluid density calculated from dirty d -waveBCS theory reproduces many of the features observed inthe experiments in Ref. 30. In particular, ρ s ( T ) showsa strong, nearly linear temperature dependence over al-most the full doping range, despite the strong suppressionof superfluid density caused by the disorder. The corre-lation between T c and ρ s [see inset of Fig. 3(b)] alsoreproduces the main features of the experiments, namelythe almost linear dependence at higher T c , with finiteintercept, crossing over to square-root behaviour at low T c . Indeed, very similar behaviour of T c ( ρ s ) was foundin earlier calculations of superfluid density for Born-limitscattering on a circular Fermi surface (Fig. 4 of Ref. 31).One concern might be that the crossover in T c ( ρ s )from linear to square-root behaviour does not occur assmoothly in the experimental data as in the theoreticalcurve. In the idealized form considered in this paper,the dirty d -wave BCS theory assumes spatial homogene-ity, whereas there is a body of evidence that real sam-ples of La − x Sr x CuO consist of an inhomogeneous mix-ture of superconducting and metallic regions, with thefraction of metallic regions increasing on overdoping. While the samples in Ref. 30 are grown with exquisitelycontrolled average composition, the doping mechanismin La − x Sr x CuO is based on random substitution ofcations and inhomogeneity must always become relevantwhen approaching the overdoped phase boundary. In ad-dition, the technique used to characterize inhomogene-ity in Ref. 30 (measurement of the temperature widthof the out-of-phase susceptibility near T c ) is not a goodprobe of in-plane microscopic inhomogeneity, as the su-perconducting coherence length diverges at T c , averag-ing over short-length-scale inhomogeneity. For an inho-mogeneous superconductor the electrodynamic responsecan be modelled using effective medium theory of theconductivity. This is described in Appendix A, wherethe macroscopic superfluid density is shown to simply bea scaled version of the microscopic superfluid density inthe superconducting regions. The effect of this type ofinhomogeneity would be to distort the theoretical T c ( ρ s )curve to the left as the inhomogeneous regime is entered,which may explain the kink at low ρ s that appears inthe experimental curve [see inset of Fig. 3(a)].Another possible concern is the degree of superfluidsuppression predicted by the calculations, which at firstsight appears surprising large. We emphasize againthat the disorder level assumed in the calculation cor-responds closely to the observed resistivity, and is re-ally the only adjustable parameter in the model. Here,a useful cross-check comes again from comparison withoverdoped Tl Ba CuO δ , where for T c ≈
25 K mate-rial the degree of superfluid suppression, ρ s /ρ s , is esti-mated to lie in the range 0.25 to 0.4, despite overdopedTl Ba CuO δ having a residual resistivity less thanhalf that of La − x Sr x CuO .Finally, while the main purpose of our calculation isto illustrate the qualitative features contained withindirty d -wave superconductivity, it is interesting thatthe implied clean-limit transition temperatures, T c , arelarge and have a suggestive similarity to the temper-atures at which the first experimental signatures ofsuperconductivity are observed in properties such asmagnetoresistance. The mean-field model consideredhere does not include fluctuation effects, which are knownto be important in the cuprates and are probably responsible for the downturns in the experimentally ob-served ρ s ( T ) on the approach to T c . Nevertheless, oneway in which the underlying T c might become visible inexperiments would be as rare regions in which the localdisorder level is lower than average. The overall impli-cation is that disorder not only plays a role in limitingsuperfluid density in the overdoped cuprates, but in sup-pressing the transition temperature. This suggests thatit may be possible to enhance T c , as well as the dopingrange over which superconductivity occurs, by controlledengineering of disorder in these materials. V. CONCLUSIONS
In conclusion, we find that dirty d -wave BCS theory,applied to a realistic parameterization of the doping-dependent Fermi surface, reproduces most of the phe-nomenology of the superfluid density in overdopedLa − x Sr x CuO . A strong suppression of superfluid den-sity is achieved without introducing significant curvaturein ρ s ( T ) by considering predominantly weak, Born-limitscattering, at a disorder level compatible with the ob-served resistivity, and in a regime that firmly satisfies theconventional definition of clean-limit superconductivity,Γ tr (cid:28) . We conclude that the correlation between T c and ρ s observed in the overdoped cuprates is a genericfeature of a disordered d -wave superconductor. ACKNOWLEDGMENTS
We acknowledge useful discussions with N. Doiron-Leyraud, P. J. Hirschfeld, F. Lalibert´e and L. Taille-fer. We gratefully acknowledge financial support fromthe Natural Science and Engineering Research Council ofCanada, the Canadian Institute for Advanced Research,and the Canadian Foundation for Innovation.
Appendix A: Effective-medium theory of aninhomogeneous superconductor
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In Section III C of our paper, the scattering parameters were reported as ρ ≈ µ Ωcm, Γ N = 18, Γ N, Born = 17and Γ N, unitarity = 1 K. The actual values used in the calculations and the preparation of the figures were a factor of π larger; i.e., ρ ≈ µ Ωcm, Γ N = 18 π , Γ N, Born = 17 π and Γ N, unitarity = π K. Note that the revised value of ρ is ingood correspondence with the residual terahertz conductivities reported in Fig. S13 of Ref. 1. In Fig. 4, the plot ofgap magnitude, ∆, showed the prefactor of the d -wave form factor cos( k x a ) − cos( k y a ) rather than the gap maximumon the Fermi surface, which is replotted below. Δ / k B T c T c (K)0 10 20 30 40 Δ / k B ( K )
10 20 30 40 50 T (K)0 FIG. 4. (Color online) Revised plot of the gap magnitude, ∆, which now correctly shows the d -wave gap maximum on theLa − x Sr x CuO tight-binding Fermi surface. Inset: Revised doping dependence of the zero-temperature gap ratio 2∆ /k B T c ,where ∆ is now the Fermi-surface gap maximum at zero temperature. Dashed line denotes the clean-limit d -wave BCS valuefor a circular Fermi surface, 2∆ = 4 . k B T c . We note that these corrections do not affect the other figures, results, and conclusions of our paper.1